Zero-temperature dynamics of Ising systems on hypercubes.

Abstract

We study the zero-temperature Glauber dynamics of homogeneous Ising ferromagnets on hypercubes, as their dimension d varies. By hypercube, we mean the d-dimensional hypercubic graph with 2^{d} vertices, where each vertex is connected to d neighbors. We investigate the asymptotic (d→∞ and time t→∞) behavior of various quantities on hypercubes, such as the final magnetization, the probability for the system to enter a ground state, etc. Our numerical studies are carried out using a uniformly random initial state but with the constraint that the initial magnetization is zero. The final states can be divided into three categories: ground states, frozen states, and blinker states. We use the notion of a k-core to describe the geometry of the frozen states and give an exponential lower bound for the number of frozen states in terms of d. Blinker states-which exist only in even d-are final states containing at least one blinker (a permanently flipping spin). Blinkers states can have rich local structures; we give explicit constructions for configurations that contain blinkers and prove that the lowest possible dimension for blinker configurations is d=8. We also study the "nature versus nurture" problem on hypercubes, asking how much the final state depends on the information contained in the initial configuration, and how much depends on the realization of the dynamical evolution. Finally, we provide several conjectures and suggest some open problems based on the numerical results.